A092681 -id:A092681 - cp's OEIS frontend

A035112 Smallest even index 2a such that n-th irregular prime p (A000928(n)) divides Bernoulli_{2a} with 0<=2a<=p-3.

32, 44, 58, 68, 24, 22, 130, 62, 84, 164, 100, 84, 20, 156, 88, 292, 280, 186, 100, 200, 382, 126, 240, 366, 196, 130, 94, 292, 400, 86, 270, 222, 52, 90, 22, 592, 522, 20, 428, 80, 236, 48, 224, 408, 628, 32, 12, 378, 290, 514, 260, 732, 220, 330, 544, 744, 102

Offset: 1

N. J. A. Sloane

nonn

The ordered pair (p(n),a(n)) where p(n) is the n-th irregular prime is called an irregular pair. Some irregular primes, such as 157, are in more than one pair. See A091887 for the number of pairs for each irregular prime. See A092681 and A092682 for higher-order irregular pairs. - T. D. Noe, Mar 03 2004

			The first irregular prime (37) divides the numerator (-7709321041217) of the 32nd Bernoulli number.

L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 350.

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A000928, A000367, A035112, A046753, A189683, A189684, A189685.

Do[ p = Prime[ n ]; k = 1; While[ 2*k < p - 3 && Mod[ Numerator[ BernoulliB[ 2*k ] ], p ] != 0, k++ ]; If[ 2*k != p - 3, Print[ 2*k ] ], { n, 3, 200} ]

More terms from Robert G. Wilson v, May 12 2001

A094095 a(n)^2 is the smallest square associated with the n-th term of A090943.

Original entry on oeis.org

103, 37, 59, 271, 37, 37, 67, 37, 59, 37, 101, 157, 37, 67, 59, 37, 37

Offset: 1

N. J. A. Sloane, May 02 2004

			103^2 divides the numerator of the Bernoulli number B(228) (a 265-digit number).

Cf. A090943, A092681.

a(4)-a(17) from T. D. Noe, May 03 2004

Name clarified by Petros Hadjicostas, May 12 2020

A090943 Even numbers n such that N(n) is divisible by a nontrivial square, say m^2 with gcd(n,m) = 1, where N(n) is the numerator of the Bernoulli number B(n). The smallest numbers m are given in A094095.

Original entry on oeis.org

228, 284, 914, 1434, 1616, 2948, 3292, 4280, 4336, 5612, 5768, 6302, 6944, 7714, 7758, 8276, 9608

Offset: 1

T. D. Noe, Feb 27 2004

This sequence consists of the union of an infinite number of arithmetic progressions. Let p be an irregular prime and let {m1, m2, ...} be even numbers < p*(p-1) such that p^2 | N(mi). Then each pair (p, mi) is a second-order irregular pair. This leads to the arithmetic progression n = mi + p*(p-1)*k for each i and for k = 0, 1, 2, 3, ... If we restrict the sequence to those pairs with mi < 10000, we find that only the pairs (37, 284), (59, 914), (67, 3292), (101, 5768), (103, 228), (157, 6302) and (271, 1434) contribute terms to this sequence.

Bernd Kellner, Über irregulaere Paare hoeherer Ordnungen [On irregular pairs of higher order], Diplomarbeit, Goettingen 2002.
S. S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers, 2018.
Charles Weibel, Algebraic K-Theory of Rings of Integers in Local and Global Fields, in: E. Friedlander and D. Grayson (eds), Handbook of K-Theory, Springer, Berlin, Heidelberg, Vol 1, 2005, pp. 139-190; see Example 96 on p. 180.

Cf. A092681, A094095.

nn=10; s = Union[284 + 36*37*Range[0, nn], 914+58*59*Range[0, nn], 3292+66*67*Range[0, nn], 5768+100*101*Range[0, nn], 228+102*103*Range[0, nn], 6302+156*157*Range[0, nn], 1434+270*271*Range[0, nn]]; Select[s, #<=10000&]

Addition of the word "smallest" in the name by Petros Hadjicostas, May 12 2020

A092682 Least number 2k such that p^3 divides the numerator of the Bernoulli number B(2k), where p is the n-th irregular prime, A000928(n).

Original entry on oeis.org

37580, 86464, 153640, 581468, 914250, 454892, 1510618, 3557642, 84974, 8905404, 11482532, 9629910, 1025814, 9252440, 6484016, 22003936, 17706562, 30054878, 18332698, 37340812, 39775150, 31082358, 5118308, 20315982, 57395934, 25079280

Offset: 1

T. D. Noe, Mar 03 2004

nonn

The ordered-pair (p(n),a(n)), where p(n) is the n-th irregular prime, is called a third-order irregular pair. Some irregular primes, such as 157, have more than one pair. See A091887 for a count of the pairs for each irregular prime.

Bernd Kellner, On irregular pairs of higher order (in German)

Cf. A000928, A035112, A091887, A092681.

cp's OEIS Frontend

A035112 Smallest even index 2a such that n-th irregular prime p (A000928(n)) divides Bernoulli_{2a} with 0<=2a<=p-3.

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A094095 a(n)^2 is the smallest square associated with the n-th term of A090943.

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A090943 Even numbers n such that N(n) is divisible by a nontrivial square, say m^2 with gcd(n,m) = 1, where N(n) is the numerator of the Bernoulli number B(n). The smallest numbers m are given in A094095.

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A092682 Least number 2k such that p^3 divides the numerator of the Bernoulli number B(2k), where p is the n-th irregular prime, A000928(n).

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